Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. They are defined as ideals such that $ab\in I$ implies $a\in I$ or $b\in I$. A maximal ideal ideal is an ideal which is maximal w.r.t. inclusion.

In the ring of integers maximal and prime ideals coincide. They are the sets that contain all the multiples of a given prime number, together with the zero ideal.

1890 questions
2
votes
1 answer

Showing $I$ in $\mathbb{Z}[\sqrt{m}]$ is or is not a maximal ideal

Suppose I have some ideal $I$ in $\mathbb{Z}[\sqrt{m}]$ (usually, $ m < 0 $). What, in general, would be a course of action to prove this ideal is maximal or prime? I know the former implies the latter, and an ideal $I$ being prime or maximal is…
tobias mulder
  • 85
1
vote
1 answer

whether $(y^2-x^3-x^2)$ is a prime ideal of $\mathbb{C}[x, y]$?

whether $(y^2-x^3-x^2)$ is a prime ideal of $\mathbb{C}[x, y]$? Many useful results for one variable (for eg polynomial ring over a field is Euclidean domain) fail in case of multiple variables. Can't guess factors of $y^2-x^3-x^2$ in $…
Vinay Deshpande
  • 779
  • 3
  • 14
1
vote
1 answer

Atiyah Macdonald commutative algebra page 4

The following is on page 4 of Atiyah and MacDonald's commutative algebra book. Example 1: Let $A=k[x_1,..,x_n]$, k a field. Let $f \in A$ be an irreducible polynomial. By unique factorization, the ideal $(f)$ is prime. Then they go on to write that…
user2371765
  • 257
1
vote
0 answers

Working out prime ideals in $\mathbb{Z}_{30}$

I am trying to figure out the prime ideals in $\mathbb{Z}_{30}$. My understanding is a little shaky so I would just like to test my understanding of the conditions for a prime ideal. So for a prime ideal $P$ in $R$ and $a, b \in R$, if $ab \in P$…
Inazuma
  • 1,103
1
vote
1 answer

Step in proof: $M$ is a maximal ideal of $R$ iff $R/M$ is a field

My book shows that: $M$ is a maximal ideal of $R$ iff $R/M$ is a field. Let $M\subset J\subset R$. They consider $R/J\cong (R/M)(J/M)$. They write $\overline R=R/M$ and $\overline J=J/M$. They claim that there is a one-to-one correspondence between…
Sha Vuklia
  • 3,960
  • 4
  • 19
  • 37
1
vote
1 answer

Find the maximum of the value $ f(x)=27\sin^8{x}+8\cos^8{x}$

Let $x\in R$,find the maximum of the value $$f(x)=27\sin^8{x}+8\cos^8{x}$$ if let $\sin^2{x}=t$,then $$f=27t^4+8(1-t)^4,t\ge 0$$ without derivative method?
math110
  • 93,304
1
vote
0 answers

Maximal ideals of $\Bbb R[x,y]/(x^2+1,y^2-1)$

My problem is finding all maximal ideals of the commutative ring $\Bbb R[x,y]/(x^{2}+1,y^{2}-1)$. I'm aware of the statement that maximal ideals of $\Bbb R[x,y]/(x^{2}+1,y^{2}-1)$ are the same as maximal ideals of $\Bbb R[x,y]$ containing…
user
  • 1,391
0
votes
1 answer

Determine which of the following ideals are prime, maximal ideal or neither in the polynomial ring $C[x,y]$

$I_{1}=(x)$, 2. $I_{2}=(x,y^{2})$, 3. $I_{3}=(x-y,x+y)$, 4. $I_{4}=(x-y,x^{2}-y^{2})$. My Attempt: It is easy to check that $I_{1}$ is prime but not maximal ideal. For $I_{2}$, clearly $y^{2} \in (x,y^{2})$, but y,y does not belong to $(x,y^{2})$.…
User124356
  • 1,617
0
votes
1 answer

$I=\{a+ib\in\mathbb Z[i]:5|a,5|b\}$ forms a maximal ideal of $\mathbb Z[i].$

How to show that $I=\{a+ib\in\mathbb Z[i]:5|a,5|b\}$ forms a maximal ideal of $\mathbb Z[i].$ I tried to prove $\mathbb Z[i]\backslash I$ is a field could not prove it.
Jave
  • 1,160
0
votes
1 answer

Prove that $(z-x^3,y-x^2)$ is a prime ideal

I am trying to prove that the ideal in ${\mathbb C}[x,y,z]$ generated by $z-x^3$ and $y-x^2$ is prime. I know I could take a suitable quotient and show it's domain. But I am rather stuck.
Math101
  • 1,106
0
votes
0 answers

Inclusion of ideals in \mathbb{C}[x,y]

I have to answer to questions. I try to give the answer myself and would appreciate if you told me if they are correct. In the ring $\mathbb{C}[x,y]$, do the following inclusions of ideals…
user689556
  • 11
0
votes
1 answer

Is $(X^2+1)$ prime ideal in $\mathbb{R}[X,Y]$?

The title already states my question: Is $(X^2+1)$ a prime ideal in $\mathbb{R}[X,Y]$? I know that if this is true, Then $\mathbb{R}[X,Y]/(X^2+1)$ must be a domain. And if that is a domain then $(X^2+1)$ must be the kernel of a homomorphism from…
jbuser430
  • 743
0
votes
1 answer

If $P$ is a maximal ideal, then $A=PP^{-1} = (1)$: Showing that $A$ is an ideal.

I'm hung up on a part of a proof for: If $P$ is a maximal ideal, then $A=PP^{-1} = (1)$. Some background: Definition of ideal I am working with is if $\alpha$ and $\beta$, both algebraic integers (roots in field $K$ of polynomials with rational…
user77970
0
votes
0 answers

maximal ideal in $\mathbb{R}$[x]

Let $\mathbb{R}$[x] be the polynomial ring over $\mathbb{R}$ in one variable .Let I$\subseteq$$\mathbb{R}$[x] be an ideal. Then 'I is a maximal ideal iff there exists a non constant polynomial f(x)$\in$I of degree $\le$ 2' Is this statement is…
dipali mali
  • 555
0
votes
0 answers

Maximal ideal and Prime ideal

Does anyone know how did the idea of maximal and prime ideal came into existence for the first time ? I want to understand their applicability in mathematics.
Dheeraj Sonker
  • 67
1
2